Ring homeomorphisms and prime ends

Abstract

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=\mu\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point $z_0\in {\overline{D}}$. In this connection, we develop the theory of the boundary behavior of the ring $Q-$homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations $\overline{\partial}f=\mu\,\partial f$, there exist regular solutions of the Dirichlet problem in arbitrary simply connected domains in $\Bbb C$ and pseudoregular and multivalent solutions in arbitrary finitely connected domains in $\Bbb C$ with boundary datum $\varphi$ that are continuous with respect to the topology of prime ends.